Exercise 12.22 [cardonforeheadexercise]
(Adapted from @Fagin+al:1995.) Consider a game played with a deck of just 8 cards, 4 aces and 4 kings. The three players, Alice, Bob, and Carlos, are dealt two cards each. Without looking at them, they place the cards on their foreheads so that the other players can see them. Then the players take turns either announcing that they know what cards are on their own forehead, thereby winning the game, or saying “I don’t know.” Everyone knows the players are truthful and are perfect at reasoning about beliefs.

Game 1. Alice and Bob have both said “I don’t know.” Carlos sees that Alice has two aces (AA) and Bob has two kings (KK). What should Carlos say? (Hint: consider all three possible cases for Carlos: AA, KK, AK.)

Describe each step of Game 1 using the notation of modal logic.

Game 2. Carlos, Alice, and Bob all said “I don’t know” on their first turn. Alice holds KK and Bob holds AK. What should Carlos say on his second turn?

Game 3. Alice, Carlos, and Bob all say “I don’t know” on their first turn, as does Alice on her second turn. Alice and Bob both hold AK. What should Carlos say?

Prove that there will always be a winner to this game.